Consider two ideal gases X and Y at a given temperature. Gas X is a...

For gas $X$ (rigid diatomic gas):

Degrees of freedom, $f_X = 3$ (translational) $+ 2$ (rotational) $= 5$

$\gamma_X = 1 + \dfrac{2}{f_X} = 1 + \dfrac{2}{5} = \dfrac{7}{5}$

For gas $Y$ (non-linear polyatomic gas with two active vibrational modes):

Each vibrational mode contributes $2$ degrees of freedom (one for kinetic energy and one for potential energy).

Degrees of freedom, $f_Y = 3$ (translational) $+ 3$ (rotational) $+ 2 \times 2$ (vibrational) $= 10$

$\gamma_Y = 1 + \dfrac{2}{f_Y} = 1 + \dfrac{2}{10} = \dfrac{6}{5}$

Ratio $\dfrac{\gamma_X}{\gamma_Y} = \dfrac{7/5}{6/5} = \dfrac{7}{6}$

Answer: $\dfrac{7}{6}$